can any one explain me full chapter lines and angles of class 9
Answers
Here are some basic definitions and properties of lines and angles in geometry. These concepts are tested in many competitive entrance exams like GMAT, GRE, CAT.
Line segment: A line segment has two end points with a definite length.
line segment
Ray: A ray has one end point and infinitely extends in one direction.
ray
Straight line: A straight line has neither starting nor end point and is of infinite length.
line segment
Acute angle: The angle that is between 0° and 90° is an acute angle, ∠A in the figure below.
acute angle
Obtuse angle: The angle that is between 90° and 180° is an obtuse angle, ∠B as shown below.
obtuse angle
Right angle: The angle that is 90° is a Right angle, ∠C as shown below.
right angle
Straight angle: The angle that is 180° is a straight angle, ∠AOB in the figure below.
Supplementary angles:
supplementary angles
In the figure above, ∠AOC + ∠COB = ∠AOB = 180°
If the sum of two angles is 180° then the angles are called supplementary angles.
Two right angles always supplement each other.
The pair of adjacent angles whose sum is a straight angle is called a linear pair.
Complementary angles:
complementary angles
∠COA + ∠AOB = 90°
If the sum of two angles is 90° then the two angles are called complementary angles.
Adjacent angles:
The angles that have a common arm and a common vertex are called adjacent angles.
In the figure above, ∠BOA and ∠AOC are adjacent angles. Their common arm is OA and common vertex is ‘O’.
Vertically opposite angles:
When two lines intersect, the angles formed opposite to each other at the point of intersection (vertex) are called vertically opposite angles.
opposite angles
In the figure above,
x and y are two intersecting lines.
∠A and ∠C make one pair of vertically opposite angles and
∠B and ∠D make another pair of vertically opposite angles.
parallel lines example 1
Solution:
Determining one pair can make it possible to find all the other angles. The following is one of the many ways to solve this question.
∠2 = 125°
∠2 = ∠4 since they are vertically opposite angles.
Therefore, ∠4 = 125°
∠4 is one of the interior angles on the same side of the transversal.
Therefore, ∠4 + ∠5 = 180°
125 + ∠5 = 180 → ∠5 = 180 – 125 = 55°
∠5 = ∠7 since vertically opposite angles.
Therefore, ∠5 = ∠7 = 55°
Note: Sometimes, the parallel property of the lines may not be mentioned in the problem statement and the lines may seem to be parallel to each other; but they may be not. It is important to determine whether two lines are parallel by verifying the angles and not by looks.
Example 2. If ∠A = 120° and ∠H = 60°. Determine if the lines are parallel.
parallel lines example 2
Solution:
Given ∠A = 120° and ∠H = 60°.
Since adjacent angles are supplementary, ∠A + ∠B = 180°
120 + ∠B = 180 → ∠B = 60°.
It is given that ∠H = 60°. We can see that ∠B and ∠H are exterior alternate angles.
When exterior alternate angles are equal, the lines are parallel.
Hence the lines p and q are parallel.
We can verify this using other angles.
If ∠H = 60°, ∠E = 120° since those two are on a straight line, they are supplementary.
Now, ∠A = ∠E = 120°. ∠A and ∠E are corresponding angles.
When corresponding angles are equal, the lines are parallel.
Likewise, we can prove using other angles too.
Example 3. If p and q are two lines parallel to each other and ∠E = 50°, find all the angles in the figure below.
parallel lines example 3
Solution:
It is given ∠E = 50°.
The two lines are parallel
→ The corresponding angles are equal.
Since ∠E and ∠A are corresponding angles, ∠A = 50° .
→ The vertically opposite angles are equal.
Since ∠A and ∠C are vertically opposite to each other, ∠C = 50°.
Since ∠E and ∠G are vertically opposite to each other, ∠G = 50°.
→ The interior angles on the same side of the transversal are supplementary.
∠E + ∠D = 180° → 50 + ∠D = 180° → ∠D = 130°
→ ∠D and ∠B are vertically opposite angles. So ∠B = 130°.
→ ∠B and ∠F are corresponding angles. So ∠F = 130°.
→ ∠F and ∠H are vertically opposite angles. So ∠H = 130°.
∠D = ∠O + 90° → 130 = ∠O + 90 → ∠O = 40°
Continue learning:
– Properties and formulas of Circles
– Types of Triangles and Properties
– Properties of Quadrilaterals (parallelograms, trapezium, rhombus)